Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics
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| Format: | Preprint |
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2024
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| _version_ | 1866909160472313856 |
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| author | Kurt, Halil Ibrahim Shen, Wenxian |
| author_facet | Kurt, Halil Ibrahim Shen, Wenxian |
| contents | The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begin{equation} \begin{cases} u_t=Δu-χ_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) ,\quad &x\in Ω\cr v_t=Δv-χ_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u),\quad &x\in Ω\cr 0=Δw-μw +νu+ λv,\quad &x\in Ω\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, and $χ_i,a_i, b_i, c_i$ ($i=1,2$) and $μ,\, ν, \, λ$ are positive constants. In [25], we proved that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution provided that $\min\{a_1,a_2\}$ is large relative to $χ_1,χ_2$, and $u_0+v_0$ is not small in the case that $(χ_1-χ_2)^2\le \max\{4χ_1,4χ_2\}$ and $u_0+v_0$ is neither small nor big in the case that $(χ_1-χ_2)^2>\max\{4χ_1,4χ_2\}$. In this paper, we proved that (0.1) has a unique positive constant solution $(u^*,v^*,w^*)$, where $$ u^*=\frac{a_1b_2-c_1a_2}{b_1b_2-c_1c_2},\quad v^*=\frac{b_1a_2-a_1c_2}{b_1b_2-c_1c_2}, \quad w^*=\fracνμu^*+\fracλμ v^*. $$ We obtain some explicit conditions on $χ_1,χ_2$ which ensure that the positive constant solution $(u^*,v^*,w^*)$ is globally stable in the sense that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0\not \equiv 0$ and $v_0\not \equiv 0$, $$ \lim_{t\to\infty}\Big(\|u(t,\cdot;u_0,v_0)-u^*\|_\infty +\|v(t,\cdot;u_0,v_0)-v^*\|_\infty+\|w(t,\cdot;u_0,v_0)-w^*\|_\infty\Big)=0. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03158 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics Kurt, Halil Ibrahim Shen, Wenxian Analysis of PDEs Dynamical Systems The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begin{equation} \begin{cases} u_t=Δu-χ_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) ,\quad &x\in Ω\cr v_t=Δv-χ_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u),\quad &x\in Ω\cr 0=Δw-μw +νu+ λv,\quad &x\in Ω\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, and $χ_i,a_i, b_i, c_i$ ($i=1,2$) and $μ,\, ν, \, λ$ are positive constants. In [25], we proved that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution provided that $\min\{a_1,a_2\}$ is large relative to $χ_1,χ_2$, and $u_0+v_0$ is not small in the case that $(χ_1-χ_2)^2\le \max\{4χ_1,4χ_2\}$ and $u_0+v_0$ is neither small nor big in the case that $(χ_1-χ_2)^2>\max\{4χ_1,4χ_2\}$. In this paper, we proved that (0.1) has a unique positive constant solution $(u^*,v^*,w^*)$, where $$ u^*=\frac{a_1b_2-c_1a_2}{b_1b_2-c_1c_2},\quad v^*=\frac{b_1a_2-a_1c_2}{b_1b_2-c_1c_2}, \quad w^*=\fracνμu^*+\fracλμ v^*. $$ We obtain some explicit conditions on $χ_1,χ_2$ which ensure that the positive constant solution $(u^*,v^*,w^*)$ is globally stable in the sense that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0\not \equiv 0$ and $v_0\not \equiv 0$, $$ \lim_{t\to\infty}\Big(\|u(t,\cdot;u_0,v_0)-u^*\|_\infty +\|v(t,\cdot;u_0,v_0)-v^*\|_\infty+\|w(t,\cdot;u_0,v_0)-w^*\|_\infty\Big)=0. $$ |
| title | Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2404.03158 |